牛顿迭代法的应用与实现

牛顿迭代法，又称为切线法，是一种高效的数值方法，广泛用于求解方程的根。其核心在于通过逐步逼近的方法，利用函数及其导数的信息来找到更精确的解。

牛顿迭代公式解析

牛顿迭代公式可以表示为：

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

其中，x_{n+1} 是下一个近似值，x_n 是当前近似值，而 f(x) 和 f'(x) 分别是目标方程及其导数。这一过程通过选择一个接近实际根的初始值（x_0) 开始，然后不断进行迭代计算。

如何选择初始值？

[燎元跃动小编]: 选择合适的初始值对于提高收敛速度至关重要。一般来说，应选取离真实根较近的位置，以便快速接近最终结果。例如，在求解方程 (x^2 - 5 = 0), 可以从 (x_0 = 2)开始。

停止条件设定

The stopping criteria for the iteration can be defined as follows:

• The absolute difference between two consecutive approximations is less than a specified tolerance.
• The number of iterations exceeds a predefined maximum limit.

[燎元跃动小编]

This ensures that the algorithm does not run indefinitely and provides an approximate solution within acceptable error bounds. For instance, in our earlier example, we can stop when the difference between two successive approximations is less than 0.001.

A Practical Example of Newton's Method

If we apply Newton's method to solve the equation (x^2 - 5 = 0), starting with an initial guess of (x_0 = 2), we perform the following iterations:

• (x_1 ≈ 2.5, ) calculated as: x_1 = x_0 - \frac{x_0^2 - 5}{(d/dx)(x^n)|_{(d/dy)}}= x_x- y_y


• (x_3 ≈  √5 ≈  ± √25 + √12 − √6/4 − √4 + ...≈ ± ...= ∞ ) stops at this point because it satisfies our stopping condition.